This file contains an algorithm for listing all cliques according to the "hybrid" algorithm of Eppstein and Strash (SEA 2011). More...

#include <limits.h>
#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include "misc.h"
#include "LinkedList.h"
#include "MemoryManager.h"
#include "degeneracy_helper.h"
#include "hybrid_algorithm.h"

Functions
int	findBestPivotNonNeighborsHybrid (int *pivotNonNeighbors, int numNonNeighbors, int vertexSets, int vertexLookup, int beginX, int beginP, int beginR, NeighborListArray *orderingArray, int scratch)
	Computes the vertex v in P union X that has the most neighbors in P, and places P \ {neighborhood of v} in an array. These are the vertices to consider adding to the partial clique during the current recursive call of the algorithm.
void	fillInPandXForRecursiveCallHybrid (int vertex, int orderNumber, int vertexSets, int vertexLookup, NeighborListArray *orderingArray, int pBeginX, int pBeginP, int pBeginR, int pNewBeginX, int pNewBeginP, int *pNewBeginR)
	Move vertex to R, set P to vertex's later neighbors and set X to vertex's earlier neighbors.
long	listAllMaximalCliquesHybrid (LinkedList adjList, int adjacencyList, LinkedList cliques, int degree, int size)
	List all maximal cliques in a given graph using the "hybrid" algorithm by Eppstein et al. (SEA 2011).
void	moveToRHybrid (int vertex, int vertexSets, int vertexLookup, NeighborListArray *orderingArray, int pBeginX, int pBeginP, int pBeginR, int pNewBeginX, int pNewBeginP, int *pNewBeginR)
	Move a vertex to the set R, and update the sets P and X.
void	moveFromRToXHybrid (int vertex, int vertexSets, int vertexLookup, int pBeginX, int pBeginP, int *pBeginR)
	Move a vertex from the set R to the set X, and update all necessary pointers and arrays of neighbors in P.
void	listAllMaximalCliquesHybridRecursive (long cliqueCount, LinkedList cliques, LinkedList partialClique, NeighborListArray orderingArray, int vertexSets, int vertexLookup, int beginX, int beginP, int beginR, int scratch)
	Recursively list all maximal cliques containing all of all vertices in R, some vertices in P and no vertices in X.

Detailed Description

This file contains an algorithm for listing all cliques according to the "hybrid" algorithm of Eppstein and Strash (SEA 2011).

Author:: Darren Strash (first name DOT last name AT gmail DOT com)

See GPL 3.0 here

See the algorithm's description in http://dx.doi.org/10.1007/978-3-642-17517-6_36

This algorithm first orders the vertices in a degeneracy order (vertices are removed from the graph in order by degree in the remaining subgraph and placed in this order in the ordering).

We then recursively call a modified version of the algorithm of Tomita et al. (2006) http://dx.doi.org/10.1016/j.tcs.2006.06.015, for each vertex v in the ordering, where R = {v}, P = v's neighbors that are after v in the degneracy order, and X = v's neighbors that are before v in the degeneracy order.

This is a recursive backtracking algorithm that maintains three sets of vertices, R, a partial clique, P, the common neighbors of vertices in R that are candidates to add to the partial clique, and X, the set of common neighbors of R that have been listed in a maximal clique with R already.

The algorithm recursively adds vertices to R from P, then updates the sets P and X to be the new common neighbors of R and recurses. When P and X are empty, R is a maximal clique, and is reported.

Updating the sets P and X is done by iterating over the later neighbors of vertices in P and X to see if they contain the vertex v added to R as a neighbor, and then testing if v's later neighbors are in P or X. Neighbors of v remain in their respective sets. For sparse graphs (graphs with low degeneracy) the number of neighbors of any vertex that come later in the degeneracy order is expected to be small. Then testing for neighbors in this way should be fast.

Besides the ordering, this algorithm includes more efficient pivot computation for sparse graphs. To find a "good" pivot, we must find the vertex in P union X that has the most neighbors in P. We accompish this step iterating through each vertex in X and P, and marking each vertex with the number of later neighbors that are in P. (if the vertex is in P, we add one to the count for the vertex itself as well) and then we iterate over the vertices has the highest count.

Function Documentation

void fillInPandXForRecursiveCallHybrid	(	int	vertex,
		int	orderNumber,
		int *	vertexSets,
		int *	vertexLookup,
		NeighborListArray **	orderingArray,
		int *	pBeginX,
		int *	pBeginP,
		int *	pBeginR,
		int *	pNewBeginX,
		int *	pNewBeginP,
		int *	pNewBeginR
	)			`[inline]`

Move vertex to R, set P to vertex's later neighbors and set X to vertex's earlier neighbors.

Parameters:

	vertex	The vertex to move to R.
	orderNumber	The position of vertex in the ordering.
	vertexSets	An array containing sets of vertices divided into sets X, P, and other.
	vertexLookup	A lookup table indexed by vertex number, storing the index of that vertex in vertexSets.
	orderingArray	A degeneracy order of the input graph.
	pBeginX	The index where set X begins in vertexSets.
	pBeginP	The index where set P begins in vertexSets.
	pBeginR	The index where set R begins in vertexSets.
	pNewBeginX	After function, contains the new index where set X begins in vertexSets after adding vertex to R.
	pNewBeginP	After function, contains the new index where set P begins in vertexSets after adding vertex to P.
	pNewBeginR	After function, contains the new index where set R begins in vertexSets after adding vertex to R.

int findBestPivotNonNeighborsHybrid	(	int **	pivotNonNeighbors,
		int *	numNonNeighbors,
		int *	vertexSets,
		int *	vertexLookup,
		int	beginX,
		int	beginP,
		int	beginR,
		NeighborListArray **	orderingArray,
		int *	scratch
	)

Computes the vertex v in P union X that has the most neighbors in P, and places P \ {neighborhood of v} in an array. These are the vertices to consider adding to the partial clique during the current recursive call of the algorithm.

Parameters:

	pivotNonNeighbors	An intially unallocated pointer, which will contain the set P \ {neighborhood of v} when this function completes.
	numNonNeighbors	A pointer to a single integer, which has been preallocated, which will contain the number of elements in pivotNonNeighbors.
	vertexSets	An array containing sets of vertices divided into sets X, P, R, and other.
	vertexLookup	A lookup table indexed by vertex number, storing the index of that vertex in vertexSets.
	beginX	The index where set X begins in vertexSets.
	beginP	The index where set P begins in vertexSets.
	beginR	The index where set R begins in vertexSets.
	orderingArray	A array storing the degeneracy order, with each vertex containing arrays of its later and earlier neighbrs in the order.
	scratch	An array that we use for scratch space to avoid allocating and deallocating memory often. We use this space to count how many neighbors each vertex has in P.

Returns:: the pivot that is chosen from P union X

long listAllMaximalCliquesHybrid	(	LinkedList **	adjList,
		int **	adjacencyList,
		LinkedList *	cliques,
		int *	degree,
		int	size
	)

List all maximal cliques in a given graph using the "hybrid" algorithm by Eppstein et al. (SEA 2011).

Parameters:

	adjList	An array of linked lists, representing the input graph in the "typical" adjacency list format.
	adjacencyList	an array of arrays, representing the input graph in a more compact and cache-friendly adjacency list format. (not currently used)
	cliques	A linked list of cliques to return. (only available when compiled with RETURN_CLIQUES_ONE_BY_ONE defined)
	degree	An array, indexed by vertex, containing the degree of that vertex. (not currently used)
	size	The number of vertices in the graph.

Returns:: the number of maximal cliques of the input graph.

void listAllMaximalCliquesHybridRecursive	(	long *	cliqueCount,
		LinkedList *	cliques,
		LinkedList *	partialClique,
		NeighborListArray **	orderingArray,
		int *	vertexSets,
		int *	vertexLookup,
		int	beginX,
		int	beginP,
		int	beginR,
		int *	scratch
	)

Recursively list all maximal cliques containing all of all vertices in R, some vertices in P and no vertices in X.

Parameters:

	cliqueCount	A pointer to the number of maximal cliques computed thus far.
	cliques	A linked list of cliques to return. (only available when compiled with RETURN_CLIQUES_ONE_BY_ONE defined)
	partialClique	A linked list storing R, the partial clique for this recursive call.
	orderingArray	A array storing the degeneracy order, with each vertex containing arrays of its later and earlier neighbrs in the order.
	vertexSets	An array containing sets of vertices divided into sets X, P, R and other.
	vertexLookup	A lookup table indexed by vertex number, storing the index of that vertex in vertexSets.
	beginX	The index where set X begins in vertexSets.
	beginP	The index where set P begins in vertexSets.
	beginR	The index where set R begins in vertexSets.
	scratch	An array that we use for scratch space to avoid allocating and deallocating memory often. We use this space to count how many neighbors each vertex has in P.

void moveFromRToXHybrid	(	int	vertex,
		int *	vertexSets,
		int *	vertexLookup,
		int *	pBeginX,
		int *	pBeginP,
		int *	pBeginR
	)			`[inline]`

Move a vertex from the set R to the set X, and update all necessary pointers and arrays of neighbors in P.

Parameters:

	vertex	The vertex to move from R to X.
	vertexSets	An array containing sets of vertices divided into sets X, P, R, and other.
	vertexLookup	A lookup table indexed by vertex number, storing the index of that vertex in vertexSets.
	pBeginX	The index where set X begins in vertexSets.
	pBeginP	The index where set P begins in vertexSets.
	pBeginR	The index where set R begins in vertexSets.

void moveToRHybrid	(	int	vertex,
		int *	vertexSets,
		int *	vertexLookup,
		NeighborListArray **	orderingArray,
		int *	pBeginX,
		int *	pBeginP,
		int *	pBeginR,
		int *	pNewBeginX,
		int *	pNewBeginP,
		int *	pNewBeginR
	)			`[inline]`

Move a vertex to the set R, and update the sets P and X.

Parameters:

	vertex	The vertex to move to R.
	vertexSets	An array containing sets of vertices divided into sets X, P, R, and other.
	vertexLookup	A lookup table indexed by vertex number, storing the index of that vertex in vertexSets.
	orderingArray	A degeneracy order of the input graph.
	pBeginX	The index where set X begins in vertexSets.
	pBeginP	The index where set P begins in vertexSets.
	pBeginR	The index where set R begins in vertexSets.
	pNewBeginX	After function, contains the new index where set X begins in vertexSets after adding vertex to R.
	pNewBeginP	After function, contains the new index where set P begins in vertexSets after adding vertex to P.
	pNewBeginR	After function, contains the new index where set R begins in vertexSets after adding vertex to R.

src/hybrid_algorithm.c File Reference

Functions

Detailed Description

Function Documentation